Chapter 2: Functions and Models Lessons 1-2: The Language of Functions and Linear Models and Correlation Mrs. Parziale

Vocabulary Univariate: one variable Bivariate: two variables relation: any set of ordered pairs independent variable: first number in an ordered pair, “x” value dependent variable: second number in an ordered pair, “y” value domain: Set of first elements (independent) range: Set of second elements (dependent)

Example 1: The base cost of renting a car is $35. There is also a charge of 40 cents per mile driven. (a)Is cost a function of miles or is miles a function of cost? (b)Independent variable is ___________________. (c)Dependent Variable is ________________ (d)Domain ________________________. (e)Range ____________________________

Example 2: Suppose in Example 1 that the car is rented for one day and the mileage is 100. (a) List three ordered pairs in the relation ___________________________________ (b) Write an equation to express the relation between miles and cost ________________

More Vocab function: a set of ordered pairs (x,y) in which each value of x is paired with exactly one value of y. function (another def): a correspondence between two sets A and B in which each element of A corresponds to exactly one element of B.

Example 3: Consider the following relation: {(1,3) (2,6) (4, 12) (-1, -3)} (a) What is the domain? ___________________ (b) What is the range? ___________________ (c) Is this relation a function? _______________

Function Notation Is a shorthand method of writing functions. (Euler’s notation) looks like this: f(x) = _____ ex. f(x) = 2x +3 x is the argument of the function.

Example 4: Consider the graph of y = x (a) What is the domain? _________________ (b) What is the range? ___________________ (c) Is it a function? ______________________

Example 5: Suppose (a) Evaluate f(1) f(2) f(3) (b) Does f(1) + f(2) = f(3)? ________________

Is the Graph a Function? Vertical Line Test: No vertical line intersects the graph of a function in more than one point. Example 6: Is each of the following a relation and/or a function?

Linear Models and Correlation This section refers to data gathered from an experiment, study, etc. If you plot each of the ordered pairs, do they come close to a line? Is the slope positive or negative?

Some Final Vocab linear function – a set of ordered pairs (x,y) that can be described by an equation of the form y = mx+b interpolation: determining a value within the domain of the x-data extrapolation: determining a value outside the domain of the data – more risky because the model may change. correlation coefficient: measures the strength of the linear relation between the two variables.

Correlation Coefficient measures the strength of the linear relation between the two variables. The sign of (r) indicates the direction of the relation between the variables positive relation – as x increases, y increases negative relation – as x increases, y decreases A perfect correlation happens when r = 1 or r = -1 Type of relations you can have between the data: Strong negativeWeak negativeNoneWeak positiveStrong positive Do not mistake correlation for causation!!

Example 7: In October 1994, Consumer Reports listed the following prices and overall ratings for drip coffee makers: ($27, 79), ($25, 77), ($60, 70), ($50, 66), ($22, 61), ($60, 61), ($35, 61), ($20, 60), ($35, 58), ($40, 54), ($22, 53), ($40, 51), ($30, 43), ($30, 35), ($20, 34), ($35, 32), ($19, 28). (a) Make a scatterplot on your calculator to determine the general relationship of the data.

(b)Find a linear model between the cost and the rating. (c)Use the model to predict the rating of a $45 coffee maker. ____________ (d) Is the prediction in part c interpolation or extrapolation? ___________________ (e)Determine the correlation coefficient (r): ____________ (f) How reliable is this model? ________________________

Try This One Age Height Make a scatterplot with your calculator. Does the data seem to be a linear relationship? What is the linear model for this data? If I want to find the height of a typical 9 year old – is this interpolating or extrapolating? How about a typical 35 year old?

Closure Given a graph or a set of data – how can you tell whether or not it represents a function? Explain how the correlation value determines the direction and strength of the correlation of a given line and set of data.